Elko under spatial rotations

TL;DR

This paper investigates the behavior of Elko spinors under spatial rotations, revealing a novel effect where they transform into superpositions of self and antiself conjugate states, a phenomenon not previously reported.

Contribution

It introduces the first analysis of Elko spinors' transformation properties under rotations, highlighting a unique superposition effect due to opposite helicity phases.

Findings

Elko spinors transform into superpositions of conjugate states under rotation.

The self/antiself conjugacy of Elko spinors is preserved despite superpositions.

This behavior differs from standard Dirac and Weyl spinors.

Abstract

Under a rotation by an angle $\vartheta$, both the right- and left- handed Weyl spinors pick up a phase factor ${\exp(\pm\, i \vartheta/2)}$. The upper sign holds for the positive helicity spinors, while the lower sign for the negative helicity spinors. For $\vartheta = 2\pi$ radians this produces the famous minus sign. However, the four-component spinors are built from a direct sum of the indicated two-component spinors. The effect of the rotation by $2\pi$ radians on the eigenspinors of the parity - that is, the Dirac spinors -- is the same as on Weyl spinors. It is because for these spinors the right- and left- transforming components have the same helicity. And the rotation induced phases, being same, factor out. But for the eigenspinors of the charge conjugation operator, i.e. Elko, the left- and right- transforming components have opposite helicities, and therefore they pick up opposite phases. As a consequence the behaviour of the eigenspinors of the charge conjugation operator (Elko) is more subtle: for $0<\vartheta<2\pi$ a self conjugate spinor becomes a linear combination of the self and antiself conjugate spinors with $\vartheta$ dependent superposition coefficients - and yet the rotation preserves the self/antiself conjugacy of these spinors! This apparently paradoxical situation is fully resolved. This new effect, to the best of our knowledge, has never been reported before. The purpose of this communication is to present this result and to correct an interpretational error of a previous version.